Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

中文翻译：

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关于具有动态Hardy型势的演化方程

受Baras和Goldstein著名论文的启发（Trans Am Math Soc 284：121–139，1984），我们研究了具有动态Hardy型奇异势能的热方程。特别是，我们对奇异点随时间变化的情况感兴趣。在适当的势和初始值条件下，我们证明了解的存在，不存在和唯一性，并且在奇异点附近获得了一个尖锐的上下界。通过使用径向热方程的解，等效积分方程的一些精确估计以及比较原理来给出证明。